Summary

From Xhost man page:

The xhost program is used to add and delete host names or user names to the list allowed to make connections to the X server. In the case of hosts, this provides a rudimentary form of privacy control and security. It is only sufficient for a workstation (single user) environment, although it does limit the worst abuses. Environments which require more sophisticated measures should implement the user-based mechanism or use the hooks in the protocol for passing other authentication data to the server.

See man xhost for the full info.

Usage

To provide access to an application running as sudo or su to the graphical server (aka your X session aka your computer screen), open a terminal and type as your normal user (don't su -):

xhost +

To get things back to normal, with controlled access to the X screen:

xhost -

The (In)famous 'cannot connect to X server :0.0' output

The above command xhost + will get you rid of that command, albeit momentarily;
one way of getting permanently rid of this issue, among many, is to add

xhost + >/dev/null

into your ~/.bashrc file. This way, each time you fire up the terminal, the command gets executed. If you do not yet have a .bashrc file in your home directory, it's OK to create one with just this line in it. If you do not add >/dev/null then each time you fire a terminal, you will see a non-disruptive message saying

access control disabled, clients can connect from any host

which is your confirmation that you can now sudo <your soft> without issue.